On differentiability and mass distributions of multivariate Archimedean copulas

Copulas, in particular Archimedean copulas are commonly viewed as analytically nice and regular objects. Motivated by a recently established result stating that the first partial derivatives of bivariate copulas can exhibit surprisingly pathological behavior, we focus on the class of d -dimensional Archimedean copulas denoted by C _ar ^d and show that partial derivatives of order (d−1) can be surprisingly irregular as well. In fact, we prove the existence of Archimedean copulas C∈C _ar ^d whose (d−1)-st order partial derivatives are pathological in the sense that for almost every x∈[0,1] ^d−1 the derivative ∂ ₁...∂ _d−1C(x,y) does not exist on a dense set of y∈(0,1). Since the existence of mixed partial derivatives of order (d−1) of a copula C is closely related to the existence of a discrete component of the corresponding Markov kernel K _C, we also study mass distributions of Archimedean copulas. Building upon the interplay between Archimedean copulas and so-called Williamson measures we show how absolute continuity, discreteness and singularity of the Williamson measure propagates to the associated Archimedean copula. Moreover, considering the Lebesgue decomposition of the Markov kernel K _C as well as the measures induced by these components and the marginal C ^1:d−1, we prove that the subclass of Archimedean copulas for which all three measures have full support, is dense in C _ar ^d. Finally, viewing C _ar ^d in the light of Baire categories, we show that, in contrast to the space of bivariate copulas, a topologically typical d -dimensional Archimedean copula C is not absolutely continuous but has a (d−1)-kernel K _C whose discrete component is degenerate, implying that pathological elements are rare in C _ar ^d.